23 research outputs found
On 2-switches and isomorphism classes
A 2-switch is an edge addition/deletion operation that changes adjacencies in
the graph while preserving the degree of each vertex. A well known result
states that graphs with the same degree sequence may be changed into each other
via sequences of 2-switches. We show that if a 2-switch changes the isomorphism
class of a graph, then it must take place in one of four configurations. We
also present a sufficient condition for a 2-switch to change the isomorphism
class of a graph. As consequences, we give a new characterization of matrogenic
graphs and determine the largest hereditary graph family whose members are all
the unique realizations (up to isomorphism) of their respective degree
sequences.Comment: 11 pages, 6 figure
On realization graphs of degree sequences
Given the degree sequence of a graph, the realization graph of is the
graph having as its vertices the labeled realizations of , with two vertices
adjacent if one realization may be obtained from the other via an
edge-switching operation. We describe a connection between Cartesian products
in realization graphs and the canonical decomposition of degree sequences
described by R.I. Tyshkevich and others. As applications, we characterize the
degree sequences whose realization graphs are triangle-free graphs or
hypercubes.Comment: 10 pages, 5 figure
On fractional realizations of graph degree sequences
We introduce fractional realizations of a graph degree sequence and a closely
associated convex polytope. Simple graph realizations correspond to a subset of
the vertices of this polytope. We describe properties of the polytope vertices
and characterize degree sequences for which each polytope vertex corresponds to
a simple graph realization. These include the degree sequences of pseudo-split
graphs, and we characterize their realizations both in terms of forbidden
subgraphs and graph structure.Comment: 18 pages, 4 figure
Split digraphs
We generalize the class of split graphs to the directed case and show that
these split digraphs can be identified from their degree sequences. The first
degree sequence characterization is an extension of the concept of splittance
to directed graphs, while the second characterization says a digraph is split
if and only if its degree sequence satisfies one of the Fulkerson inequalities
(which determine when an integer-pair sequence is digraphic) with equality.Comment: 14 pages, 2 figures; Accepted author manuscript (AAM) versio